Category Archives: 2.1 Curriculum

“I’ve gotten this far… I may as well finish it”

With the amount of times it has happened already, I should not really be surprised that the next area covered in Discovering Mathematics is another one that not only astounds me, but applies to me directly – and yet, my jaw is once again left near-enough dropped as I began to uncover Counter Intuitive Mathematics.

Story time – there’s a point I want to make, I promise. A few years back now in 2013, I downloaded an app from the App Store called ‘The Simpsons: Tapped Out’, which would turn into an addiction I would hold onto for some time after.

The premise of the game was fairly simple – you complete quests, build your ‘Springfield’ and level up. Special events would be held throughout (for example, at Halloween or Christmas) where for a restricted amount of time, there were limited items in the game you could unlock, should you manage to collect enough of the event currency. Like a lot of things, it was fun at first. Quite quickly, I became quite invested in the game and it became very much a part of my routine in a desperate frenzy to get every item possible. Eventually though, after spending a few years playing the game, I was ranking up my event currency from the latest update for the next prize when I began to reach a disheartening conclusion. This wasn’t enjoyable anymore – at all. All I was doing, was tapping for the sake of completion. Tap, gain points, tap gain points, and so on. I was not having fun, yet for some reason, I kept on playing. I continued on, with my mind thinking back to all the time (and real money at points!) I had spent on this game, and how I didn’t want any of it to go to waste. The game became much more a chore than a hobby, which is never enjoyable. In this scenario, there was no end.

Although the game is actually still going somewhat strong today (Musgrave, 2018), after several guilty return trips to the game, I did eventually leave the time-consuming annoyance behind. This experience however, is a personal example of the Sunk Cost Fallacy. I had a difficult time letting go of a past cost which could not be recovered. I felt defeated.

The video below features Julia Galef, who explains the Sunken Cost Fallecy in a relatable, and therefore understandable sense:

The decision to leave the game behind was a tough one, having accumulated so much and spent so much of my time and effort into. I had to make the decision that was best for me, and that meant slashing out the idea that should I continue to play, I would be gaining a non-existent hypothetical value. This links strongly to what is referred to as ‘Loss Aversion’.

 

As humans, we would much rather avoid losing than losing. Obviously. What is more interesting however, is that studies have shown that we would much rather avoid losing than winning (Kay, no date). For example, if I was to wake up one morning and find a £50 note in my driveway, I would be delighted and filled with excitement. If however, I was to come home one day after work and realise that I had lost an existing £50 note, my feelings of dismay and frustration would outweigh the former outcome.

 

Along with the Sunk Cost Fallacy, Kruger, Mirtz and Miller (2005) reference Loss Aversion as a reason to changing our answers. They refute against the expressions ‘go with your gut’ or ‘stick with your instincts’ that others such as Brownstein, Wolf and Green (2000) have made. They claim that that the majority of decision changes result in a win/correct answer. ‘The Monty Hall problem’ supports this statement with some admittedly at first glance perplexing, but fascinating mathematics.

Scenario – a gameshow tells you to pick a door. One has a car. The other two have goats. Whatever option you choose (A, B or C), the host will always show you where ONE of the other goats are. At this point, you are asked to stick or switch. Without analysis, it would appear that since one goat will be revealed and discarded, you now have a 50% chance of winning or losing either way – but this is incorrect! (Mitzenmacher, 1986).

Taken from Wikipedia

In fact, you have a much higher chance of winning the car if you switch. What took me time to get my head around personally was the fact that the odds do not change from 33.3(333333…..)% to 50%. They in fact shift over the remaining closed door (Mitzenmacher, 1986). Of course, a win is not guaranteed, but it is certainly more likely. On the well-established YouTube channel Numberphile, you can find a video which explains this in a lot more detail. Click here.

With similar links to my previous post on probability (which you can view here), we can take the basic concepts of percentages and chance and apply it to a much grander situation. This links into the idea of connectedness, one of four key properties that Liping Ma (2010) underlines as what an individual needs to have for a Profound Understanding of Fundamental Mathematics (PUFM). The options listed in the Monty Hall Problem feeds the second property, multiple perspectives – there are advantages and disadvantages for any outcome chosen, though the former will probably prevail over the latter due to the nature of the predicament. Likewise, different approaches can be made when it comes to The Sunk Cost Fallacy. That is, to continue on with something for your own perceived satisfaction, or leave it behind for the greater good – both of which at a cost of losing something (e.g. time money). The basic idea of loss aversion of course can have either small or catastrophic consequences – this is something recognised by Barclay’s in reference to investments.

Would it be fair to say that Maths makes our decisions for us? Of course not. But can it influence them? Most definitely. A PUFM will be extremely beneficial in relation to this aspect. From a personal standpoint, given the repetitive nature of The Simpsons Tapped Out, I am willing to bet I wasn’t the only one who had this experience. However, I am also willing to bet that some people may not have realised the process they are going through, or if they do, will fail to escape in an attempt to avoid perceived loss.

References

Big Think (2013) Julia Galef: The Sunk Cost Fallacy. Available at: https://www.youtube.com/watch?v=vpnxd31y0Fo (Accessed: 3 November 2018)

Brownstein, S., Wolf, I., and Green, S. (2000) Barron’s ‘How to Prepare for the GRE: Graduate Record Examination’. (14th edn.) New York: Barrons Educational Series Inc.

Kay, M. (no date) You can implement these tips to your site or you can keep losing subscribers every day. The story of loss aversion. Available at: http://psychologyformarketers.com/loss-aversion/ (Accessed: 3 November 2018)

Kruger, J., Wirtz, D., & Miller, D. T. (2005). Counterfactual Thinking and the First Instinct Fallacy. Journal of Personality and Social Psychology, 88(5), 725-735. Available at: http://dx.doi.org.libezproxy.dundee.ac.uk/10.1037/0022-3514.88.5.725 (Accessed: 3 November 2018)

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. (Anniversary Ed.) New York: Routledge.

MindfulThinks (2017) Sunk Cost Fallacy And Why You Should Quit. Available at: https://www.youtube.com/watch?v=xXpzfy5oKWg (Accessed: 3 November 2018)

Mitzenmacher, M (1986) The Monty Hall Problem: A Study. Available at: https://pdfs.semanticscholar.org/4ece/81b2830bb47e279920f7cf92d672ba5a1373.pdf (Accessed: 3 November 2018)

Musgrave, S. (2018) Best iPhone Game Updates: ‘Marvel Contest of Champions’, ‘Mines of Mars’, ‘Temple Run 2’, ‘Choice of Games’, and More. Available at: https://toucharcade.com/2018/10/08/best-iphone-game-updates-marvel-contest-of-champions-mines-of-mars-temple-run-2-choice-of-games-and-more/ (Accessed: 3 November 2018)

mybarclayswealth (2016) Loss Aversion. Available at: https://www.youtube.com/watch?v=OfQkGoNS15Q (Accessed: 3 November 2018)

Images Used

https://en.wikipedia.org/wiki/The_Simpsons:_Tapped_Out

https://deepvalueetfaccumulator.com/behavioral-finance-and-the-value-investor/consequences-panic-selling-almost-always-bad/attachment/loss-aversion/

https://math.stackexchange.com/questions/1684694/could-the-monty-hall-problem-be-applied-to-multiple-choice-tests

Chance of Me Gaining an Appreciation for Maths – Extremely Likely

Probability – the next (and successful) leap in convincing me of how relevant mathematics actually is. This module has been effective in changing previously negative perceptions of maths, one of which in particular is that the terms ‘maths’ and ‘complex’ go together like bread and butter – this is not the case. Granted, parts of it can, as with anything. However I can vouch from my personal ongoing experience that if if we allow it to, it can be an enjoyable art.

Very briefly, probability is just what it sounds like – the likelihood of a certain event taking place or not. (Boaler, 2009). To use one of the most basic examples, the likelihood of rolling a ‘4’ on a dice. Well, there’s only 1 ‘4’ on a dice and this is known as the sample point. There are 6 possible outcomes overall (the values of 1-6), and this is known as our sample space. The probability of ending up with a ‘4’ on the roll of a dice is 1/6. Whilst the terminology was new to me, the concept is simple enough (Don’t Memorise, 2014).

Image result for probability

Schools Minister Nick Gibb identified 3 purposes to education in a 2015 speech, one of which was “preparation for adult life” – a quotation that is very fitting in regards to probability (The Purpose of Education, 2015). Like many other aspects of mathematics such as money and time, chance and probability is a relevant concept we use on a daily basis. The weather is an excellent example of this assertion. Before leaving for the bus in the morning, it’s very likely I’ll ask “Alexa, what is the weather in Dundee today?”. Now personification aside, I would be very surprised if she told me “It will definitely rain today for 3 hours straight”.

What she might tell me is “It probably won’t rain in Dundee today. There’s only a 25% chance“.

Since meteorologists cannot predict the exact weather conditions, they must make informed predictions (Tucker, 2018). The information we take from them may not emerge as being correct however – a 25% chance of rain does not eliminate the possibility. Regardless, it does help us make Image result for monopoly manjudgement calls – will I take a raincoat, or can I leave it in my back so there’s more space for collecting my library books? With only a 25% chance, its probably going to be the latter of the two. The decisions we make from probability do not just have to relate solely the weather, of course (Haylock, 2006). Informed decisions are a major part of any human’s life, and they can range from purchasing properties in Monopoly at a family game night, to choosing the best day to practice surfing based on the tidal/wave conditions, or indeed, making the right call in the world of gambling.

I myself work at an indoor amusements/arcade, however my responsibilities of managing children’s parties mean I have never dabbled with the ‘nudgers’ that to me are nothing more than complex machines with countless buttons and symbols on them. Unlike the simple example of rolling a singular die to get a number explained above, these machines have multiple possible combinations (i.e. the sample space).  Obviously, as with any business, the company intends to make a profit from the service it provides. As such, they are going to benefit in winnings a lot more than we are – even if we don’t witness that fact. The video below explains the basic functions of a slot machines, including what is meant by the terms theoretical payout and actual payout.

With these terms in mind, I spoke to one of our regular customers, who makes a guaranteed (okay, ‘extremely likely’) appearance at my work every weekend. I questioned him about his knowledge of gambling, and was consequently given a tour of the arcade, where he impressed me by pointing out which machines are best for winning (i.e. the best turnout rate, along with which ones were likely to give you an extra pound coin that may be stuck in the coin mechanism, or which machine is likely to have some loose change hidden at the back of the darkened retrieval pot. This man in particular has been coming to the establishment for years, so I was very interested to know how much profit his knowledge and skill set have rewarded him with – except he doesn’t. To lift his own words, “pure entertainment value”. He knows he is going to lose money regardless of what he plays, and this once again links back to how the actual payout of slot machines are ALWAYS greater than the money put in – you are technically losing more than getting back (tech4truth, 2010). This is not a new concept either. Charles Fey, who invented the first slot machine (3 reels, 5 symbols), had a 75% average pay-back! That may seem great at first glance, it only paid out 50% of the time (Valentine, 2018). There may be that faulty machine in the corner of the room, or perhaps you are being watched by a ‘lurker’ waiting to pounce at the right opportunity to steal your winnings. Even then though, they are using their knowledge of probability of knowing when to swoop in.

Image result for PHOEBE AND THE LURKER

To summarise all of this, probability and chance is something we use everyday in a variety of circumstances. It can become part of subconscious, much like the process of respiration and blinking, where we only realise we are doing it when we are told so. This area in particular is such a huge foundation in regards to our everyday choices. I’m truly beginning to realise that Maths isn’t just about sitting at a high school desk, attempting to find ‘x’ and losing your mind whilst you do it. There’s so much more to it, and just because it’s not an equation, that doesn’t make it any less maths-related.

Also, apologies – chances are you’re now aware you’re breathing and it won’t be a subconscious process for a little bit.

 

References

Boaler, J. (2009) The Elephant in the Classroom: Helping Children to Learn and Love Maths.  London: Souvenir Press Ltd.

Don’t Memorise (2014). Probability – Sample Space, Sample Points, Events! Available at: https://www.youtube.com/watch?v=5oI8-iQqPAI (Accessed: 18th October 2018)

Gambling Glossary: A Guide to Gambling Terms. (no date). Available at: https://www.gambling.net/glossary.php (Accessed: 18th October 2018)

Haylock, D. (2006) Mathematics explained for primary teachers. London: SAGE.

Probability. (no date) Available at: https://www.mathsisfun.com/data/probability.html (Accessed: 18th October 2018)

tech4truth (2010) Slot Machine Paybacks and Slot Odds Explained (Tech4Truth Episode 3) Available at: https://www.youtube.com/watch?v=4wzg-8QKC5s (Accessed: 18th October 2018)

The purpose of education. (2015) Available at: https://www.gov.uk/government/speeches/the-purpose-of-education. (Accessed: 18th October 2018)

Tucker, K. (2018) Examples of Real Life Probability. https://sciencing.com/examples-of-real-life-probability-12746354.html (Accessed: 18th October 2018)

Valentine, E. (2018). ‘Chance and probability’ [PowerPoint presentation]. ED21006: Discovering Mathematics. Available at: https://my.dundee.ac.uk/webapps/blackboard/execute/displayLearningUnit?course_id=_58988_1&content_id=_5217952_1. (Accessed: 18th October 2018)

Images Used

https://www.wikihow.com/Calculate-Probability

https://www.pinterest.co.uk/pin/330873903853107243/?lp=true

https://www.pinterest.co.uk/pin/451134087656430346/?lp=true

Time + Commitment + Understanding = Success

Let me start off my saying the word ‘yes’. Yes, I am aware that we use maths everyday, whether that be in telling the time so you are not to miss your train, or working out the change you are due when paying for your bus early on a Monday morning. Whilst I am not in a place, nor do I want to be in a place, to speak for others, I find this obvious. Of course, we use maths everyday, even if we don’t necessarily realise it at the time of putting it into practice.

Initially, I am then left pondering the question “What exactly am I going to learn in this class?”. The module is titled ‘Discovering Mathematics’. In aid of finding out an answer, looked back on my personal experience of Mathematics – if one could call it that.

More so in the later stages of primary school, the most vivid memories I can recall of my maths ‘lessons’ was using the teacher’s best friend – TJ Textbooks. Complete a page, get it marked. Complete a page, get it marked. Complete a page, get it marked. That format changed very Image result for tj publishersrarely, and when it did, it usually followed the similar format of ‘complete the end of chapter assessment, get it marked’. Hence, going on placement last semester and seeing the range of ways Maths was taught was a bit of shock delight. One group having a lesson with the teacher on the carpet, one group digitally learning on the iPads, one group working independently, one group working through activities with the Teaching Assistant… but I digress. A variety of ways to support ones learning and understanding of number work was used on placement. For me personally, this was not the case. We were moving on topic by topic regardless. If you didn’t get it, maybe you would get moved down a group, where you would repeat the same work at a later date with the assumption that since you have been pushed back, the work is now doable.

Secondary school followed a similar experience, particularly in undertaking National 5 and Higher classes. There was so simply no time, in the teachers eyes, to flesh out the most interesting and engaging Maths lessons possible. Their prime focus, and rightfully so in my opinion given the circumstances, was to get us our qualifications. The teachers are at no real fault here. They have been given a deadline, which I experienced many times over, of content that needs to be in our heads by May. If it’s not, we fail. They don’t have the time to let us fully comprehend logarithms – partially evidenced by the fact that I have absolutely no idea in regards to their function or place in life outside of the Maths classroom, despite undertaking the Higher course twice.

My teacher, whom I will refer to us as Mrs. Says-It-How-It-Is, admitted to me very early on in 5th year (my first attempt of the Higher course) that she was fairly certain that me being able to cram enough of this content into my head with around 6 months until the exam was going to be an extremely difficult task. I remember being in-denial of this at the time, with the optimist inside me fighting through the course. I attended every study support possible, took my poor prelim score with a pinch of salt, worked tirelessly on past papers – yet I failed. I was devastated. What now? If my teacher knew I was going to fail since last Autumn, is anything achievable if someone already believes it’s a fool’s errand?

Image result for failure

Fast forward a year however and my facial expression of sadness and despair had turned right side up. I passed, with a grade B nonetheless. Upon re-visiting the school to thank my teacher, the simple and casual (to her) but powerful (to me) words she spoke to me have stuck with me ever since, and this module is an excellent place to shed some light on them:

“Some people just need a little big longer. For some, they need the two years.”

Okay… so what? Who cares, and why is this relevant? It’s relevant because of the message I took from it, and this relates to the importance of time allowance.

Liping Ma (pictured below) writes of four fundamental principles of mathematics:

  1. “Inter connectedness”Image result for maths liping ma
  2. “Multiple perspectives”
  3. “Basic ideas/principles”
  4. “Longitudinal coherence”

Like many subjects, especially Language, we are determined to inject the children with as much information as possible from the get-go. This is not to say building strong foundations isn’t important, however we get extremely worried if they are not reading to our expectations, or cannot recite their 8 times table off by heart at a young age.

Linking back into the four principles, if one is to properly adhere to them then slowing things down is one the best ways to do that. Let our classes have the time to understand the relationships between the different basic principles. Allow them to understand the role something plays in a mathematical process. Make it a mission for the youths to comprehend how something develops overtime. If we can’t do this, then there is an issue of the introduction of misconceptions and confusion. We as Primary Teachers are vital in perpetuating these ideas – because for the time being, secondary school staff (specifically those teaching senior school) simply can’t – quite literally.

As a second year student, I am far from knowing all the answers. Very far, in fact. Luckily however, I now have the one key thing I was babbling on about above – time. The Discovering Maths module is now a place of hope where I can not only get over my anxiety of maths, but really get to grips at how vital it is in everyday life – because I am willing to bet it’s far more than handling money and following an itinerary.  I now have a safe space to truly explore the diverse subject and hopefully gain an understanding of how on earth I am going be teaching it for the rest of my life.

Images used:

https://www.amazon.co.uk/TeeJay-5-14-Maths-Level-D/dp/1907789049

https://beinspiredchannel.com/handle-failure/

https://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma.html

This blog post also references the first input on the Discovering Maths module given on 10/09/18 by Jonathan Brown. 

Author’s Note: Discovering Maths 1

 

Providing a Structure to Eliminate Fear – The Drama Lesson

Drama is one of the curricular areas in which many teachers fear teaching. They don’t want to take the risk that if they bring their class onto the school stage, chaos will erupt, and behavioural management becomes more difficult to implement than ever before. With these assumptions, I ask how are we to encourage children to take risks in their learning if we are not doing so ourselves? Dickenson and Neelands write about how it could convey the need for gender mixing and space and resource management.

This video provides a form of solution that teacher’s may exhibit when planning the drama lesson, and actually turn it round into one of the most exciting moments of the timetable.

Image result for drama

The first step is establishing an agreement The Drama lesson should always begin with an agreement. This is a necessity, as no individual can be forced into participating. As such, a negotiation can be put in place to encourage joining in. For those pupils who might struggle to do so, using drama techniques in the classroom may be an introductory method to aid them e.g. hot seating. According to Dickenson and Neelands, there are no shortcuts to strategy management, and the issues that need solved are likely to be magnified within Drama.The lesson conveners in the video use the ‘Three C’s’. If a problem were to occur, it’s likely the child facing it is either not communicating effectively, co-operation fairly, or concentrating fully. As teachers, we must be mindful that everyone in the Drama lesson must follow the agreed contract, and that when said contract is broken, everyone should be responsible for deciding the consequence. Consistency is key. Otherwise, what is the point of the contract being in place?  Warming up is obviously where the body is prepared for the forthcoming activity. What perhaps isn’t as glaring is that the mind is also being prepared, as the participants are beginning to differentiate from the Maths lesson they just had in the classroom 15 minutes ago. The video referenced above then conveys the importance of given the children a focusto perpetuate further development. Photographs were used as a stimulus to establish a clear focus. As the children take grasp of this, an allowance for deeper thinking of initial ideas is provided. Visualisation is where the teacher tells a story as children close their eyes, translating the teacher’s words into an imaginative picture in their minds. The soundscape is where the pupils become engrossed into their environment and they can begin to share what they hear and where these noises are originating from. This can link to other areas of the curriculum e.g. geography: the different features found in various environments. The bodyscape then allows pupils to create their own structure, using nothing apart from their own and their peers bodies. No props – but still lots of room for improvisation. As the room/hall falls silent, a simple tap on the shoulder from the teacher allows the pupils to voice their feeling: what are they thinking? After a lesson of active learning (not that active has to be moving around!), it’s vital that there is time and effort made to provide an evaluation, as a calming from the physical activity. This segment may include, though not visible, a self realisation for the children. During Drama, all strengths of a given group must be used, and so children may begin to rethink their opinions on their peers.

Image result for drama

References – all accessed 03/02/18

Text Content:
Dickenson, R & Neelands, J. (2006) Improve Your Primary School Through Drama. Oxon: David Fulton Publishers. – Chp.3: Getting Ready for Drama.

http://archive.teachfind.com/ttv/www.teachers.tv/videos/ks1-ks2-drama-teaching-drama-a-structured-approach.html

Images:

https://commons.wikimedia.org/wiki/File:No_drama.svg

https://commons.wikimedia.org/wiki/File:Comedy_and_tragedy_masks_without_background.svg